论文标题
凸形的组
Groups of convex bodies
论文作者
论文摘要
在本文中，我们介绍和研究了一个拓扑的Abelian凸体群，类似于剪刀一致性组和McMullen的多层代数，其通用性能与凸面上的持续价值相对应，涉及凸面上的持续同质性。为了研究这一组，我们首先获得了McMullen多项式的版本，以估计不在田地或向量空间而是在Abelian组中获得值。使用此功能，我们能够为凸体组配备一个由所有正度的真实矢量空间组成的分级，从而反映了多层代数的主要结构特性之一。希望这项工作能够成为对凸面上估值的k理论解释的起点。
In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen's polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mirroring one of the main structural properties of the polytope algebra. It is hoped that this work can serve as the starting point for a K-theoretic interpretation of valuations on convex bodies.
